How low-cost AI universal ap proxima tors reshape market efficiency
低成本AI通用逼近器如何重塑市场效率
Abstract
摘要
The efficient market hypothesis (EMH) famously stated that prices fully reflect the information available to traders [Fama, 1970]. This critically depends on the transfer of information into prices through trading strategies. Traders optimise their strategy with models of increasing complexity that identify the relationship between information and profitable trades more and more accurately. Under specific conditions, the increased availability of low-cost universal ap proxima tors, such as AI systems, should be naturally pushing towards more advanced trading strategies, potentially making it harder and harder for inefficient traders to profit. In this paper, we leverage on a generalised notion of market efficiency, based on the definition of an equilibrium price process, that allows us to distinguish different levels of model complexity through investors’ beliefs, and trading strategies optimisation, and discuss the relationship between AI-powered trading and the time-evolution of market efficiency. Finally, we outline the need for and the challenge of describing out-ofequilibrium market dynamics in an adaptive multi-agent environment.
有效市场假说 (EMH) 著名地指出，价格完全反映了交易者可获得的信息 [Fama, 1970]。这关键取决于通过交易策略将信息转化为价格。交易者通过越来越复杂的模型优化他们的策略，这些模型越来越准确地识别信息与盈利交易之间的关系。在特定条件下，低成本通用近似器（如 AI 系统）的可用性增加，自然会推动更先进的交易策略的发展，可能使低效交易者越来越难以获利。在本文中，我们基于均衡价格过程的定义，利用广义的市场效率概念，使我们能够通过投资者的信念和交易策略优化来区分不同层次的模型复杂性，并讨论 AI 驱动的交易与市场效率时间演变之间的关系。最后，我们概述了在自适应多智能体环境中描述非均衡市场动态的需求和挑战。
1 Introduction
1 引言
According to many economists markets are the most efficient mechanism for collecting, aggregating, and condensing widely diverse information into a single estimate expressed by the price [Black, 1986]. Markets do that with trading. In financial markets, traders collectively shape price dynamics by mapping their information into trades. If many traders agree that an asset is worth more than what it is selling for, they will try to buy it, thus collectively pushing the price up and eliminating the potential arbitrage. In doing so, traders’ individual judgements, and the information those judgements are based on, become ‘priced in the market’. Thus, the process of seeking a profit ends up enhancing the efficiency of financial markets.
根据许多经济学家的观点，市场是收集、汇总和浓缩多样化信息的最有效机制，这些信息最终通过价格表达出来 [Black, 1986]。市场通过交易实现这一功能。在金融市场中，交易者通过将他们的信息映射到交易中，共同塑造价格动态。如果许多交易者认为某项资产的价值高于其当前售价，他们会试图购买它，从而共同推高价格并消除潜在的套利机会。在此过程中，交易者的个人判断以及这些判断所基于的信息被“定价到市场中”。因此，追求利润的过程最终提高了金融市场的效率。
The theory of market efficiency, [Fama, 1970], in its original strong formulation, hypothesizes that prices reflect all the available information at all times and that this should imply that price changes will be independent and identically distributed, and under specific conditions follow a random walk, [Bachelier, 1900, Samuelson, 2016]. This original hypothesis did not explain what happens when new information is incorporated, as it implied an instantaneous transmission of new information into price changes. This information transmission is paradoxical: price changes only occur as a consequence of a trade, a trade that should be based on new information not reflected in the price until it occurs. Hence, informed trades should exist in order for the price to adjust to new information,[Grossman and Stiglitz, 1980].
市场效率理论 [Fama, 1970] 在其最初的强形式中假设价格始终反映所有可用信息，这意味着价格变化应该是独立且同分布的，并且在特定条件下遵循随机游走 [Bachelier, 1900, Samuelson, 2016]。这一原始假设并未解释新信息被纳入时会发生什么，因为它暗示了新信息会瞬时传递到价格变化中。这种信息传递是矛盾的：价格变化仅作为交易的结果发生，而交易应基于尚未反映在价格中的新信息。因此，为了使价格调整到新信息，知情交易必须存在 [Grossman and Stiglitz, 1980]。
Despite its paradoxical nature, the theory of efficient markets does capture a fundamental understanding of market functioning in finance, as it gives a framework to discuss the connection between the (im)possibility of traders to make a profit based on the available information and the properties of price time series. For example, as shown in [Delbaen and Sch a cher mayer, 1994], the absence of arbitrage opportunities should lead to the existence of an equivalent local martingale measure for the price process.
尽管有效市场理论具有矛盾性，但它确实捕捉到了金融市场运作的基本理解，因为它提供了一个框架来讨论交易者基于可用信息获利的（不）可能性与价格时间序列特性之间的联系。例如，如 [Delbaen and Schachermayer, 1994] 所示，无套利机会应导致价格过程存在一个等价局部鞅测度。
The different original formulations of the theory of market efficiency, [Fama, 1970], distinguish between discrete levels of information, i.e. public and private, but do not clarify what are the implications for a trader operating in mixed marketplaces where information is unequally distributed and traders use models with different levels of intelligence, i.e. with a different ability to extract signal from a given information set. Furthermore, the hypothesis is largely independent on the capability of traders to make new trades, due to endogenous constraints, such as dry powder, or on the market power and potential market impact of the trades of large investors, but also exogenous factors, such as high yield rates, and thus putting the ability to trade on an equal footing with the willingness to trade.
市场效率理论的不同原始表述 [Fama, 1970] 区分了离散的信息层次，即公开信息和私人信息，但并未阐明在信息分布不均且交易者使用不同智能水平的模型（即从给定信息集中提取信号的能力不同）的混合市场中，交易者操作的含义。此外，该假设在很大程度上独立于交易者进行新交易的能力，这可能是由于内生约束（如可用资金）或大投资者交易的市场力量和潜在市场影响，但也包括外生因素（如高收益率），从而将交易能力与交易意愿置于同等地位。
In this paper, we consider recent advances in the understanding of market efficiency, such as [Timmermann and Granger, 2004] and [Jarrow and Larsson, 2012], which account for trading strategies, information asymmetry and model complexity, and use it to formulate a series of hypotheses on the expected evolution of (in)efficient markets in presence of a growing fraction of AI traders. Previous works have discussed, also critically, the impact of the whole algorithmic trading and automation on market efficiency, [Yadav, 2015], and only recently scholars have directly addressed the question of the specific role of AI in shaping market efficiency [Marwala et al., 2017, Mancuso, 2022]. Here, leveraging on a consolidated mathematical framework used to model market efficiency, we are able to discuss different levels of market efficiency and discuss why low-cost universal approximator, such as AI systems, constitute a potential novelty in the evolution of market efficiency. We hypothesize that the deployment of AI models which are able to train larger and larger classes of models at lower optimisation costs should force traders to follow higher standards of efficiency while minimizing the use of human-defined trading strategies.
在本文中，我们考虑了市场效率理解的最新进展，例如 [Timmermann and Granger, 2004] 和 [Jarrow and Larsson, 2012]，这些研究考虑了交易策略、信息不对称和模型复杂性，并利用这些进展来制定一系列关于在 AI 交易者比例不断增加的情况下（非）有效市场预期演变的假设。先前的研究也批判性地讨论了整个算法交易和自动化对市场效率的影响 [Yadav, 2015]，而直到最近，学者们才直接探讨了 AI 在塑造市场效率中的具体作用 [Marwala et al., 2017, Mancuso, 2022]。在这里，我们利用一个用于建模市场效率的成熟数学框架，能够讨论不同层次的市场效率，并解释为什么低成本通用逼近器（如 AI 系统）构成了市场效率演变中的潜在新事物。我们假设，部署能够以较低优化成本训练越来越多种类模型的 AI 模型，应该迫使交易者遵循更高的效率标准，同时尽量减少对人类定义的交易策略的使用。
Finally, we discuss the role of population dynamics in the emergence of market efficiency and the necessity of detailed agent-based models and game-theoretical analysis to understand the functioning of financial markets, especially in the presence of AI-human interactions.
最后，我们讨论了人口动态在市场效率出现中的作用，以及基于详细智能体模型和博弈论分析的必要性，以理解金融市场的运作，特别是在存在AI与人类互动的情况下。
2 Literature review
2 文献综述
Market efficiency is a fundamental concept in both finance and economics and has been investigated from many perspectives and with different levels of mathematical rigor. In this study, we mainly focus on the direct ramifications of the seminal work on the efficient market hypothesis by Fama [Fama, 1970], which focuses on the relationship between the information available to traders, their ability to make predictable profits, and how this is reflected in the statistical properties of financial returns.
市场效率是金融学和经济学中的一个基本概念，已从多个角度并以不同的数学严谨性进行了研究。在本研究中，我们主要关注 Fama [Fama, 1970] 关于有效市场假说的开创性工作的直接影响，该假说关注的是交易者可获得的信息、他们获得可预测利润的能力以及这些如何反映在金融回报的统计特性之间的关系。
Famously, the efficient market hypothesis has three formulations: the weak, semi-strong, and strong one. In the weak formulation, the statement is the following: no trader is able to make a predictable profit solely based on available public market information. Its semi-strong formulation reads: no trader is able to make a predictable profit based on all the available public information (market and beyond). Finally, its strong formulation states: no trader is able to make a predictable profit based on both public and private information, i.e. no predictably profitable trading strategy is possible in a strongly efficient market. In other words, [Fama, 1970], “The strong form tests of the efficient markets model are concerned with whether all available information is fully reflected in prices in the sense that no individual has higher expected trading profits than others because he has monopolistic access to some information.” This notion was initially, and still is at times, considered as an indication that neither technical nor fundamental analysis or even insider trading, can allow traders to achieve returns greater than those that could be obtained by a random buy and hold portfolio with the same level of risk, [Malkiel, 1989].
著名的有效市场假说有三种表述：弱式、半强式和强式。在弱式表述中，其陈述如下：没有任何交易者能够仅基于可用的公开市场信息获得可预测的利润。其半强式表述为：没有任何交易者能够基于所有可用的公开信息（市场及其他）获得可预测的利润。最后，其强式表述为：没有任何交易者能够基于公开和私人信息获得可预测的利润，即在强式有效市场中，不存在可预测的盈利交易策略。换句话说，[Fama, 1970] 指出：“有效市场模型的强式检验关注的是所有可用信息是否完全反映在价格中，即没有任何个人因为垄断了某些信息而拥有比其他交易者更高的预期交易利润。”这一概念最初，并且有时仍然被认为表明，无论是技术分析、基本面分析，甚至是内幕交易，都无法让交易者获得比具有相同风险水平的随机买入并持有组合更高的回报，[Malkiel, 1989]。
Market efficiency, as a blanket concept that hypothesizes collective forecasting consequences on price time series due to the strategies of traders, was expanded and adapted by [Timmermann and Granger, 2004]. In this work, the authors introduced a model-specific definition of market efficiency, where price changes are independent and unpredictable conditioned on the forecasting model that the traders are using to map the available information into predictions: in an efficient market there exists an entire set of models for which the expectation of future discounted return is zero. Moreover, the authors introduce a notion of time locality in market efficiency, i.e. the possibility for markets to be temporarily inefficient and then become efficient, or the other way around. Indeed the evolutionary dynamics of the market participants and of their trading strategies reflects the adaptive nature of market efficiency, [Lo, 2005], and poses the challenge of modelling out-of-equilibrium market dynamics, i.e. how financial markets approach efficiency, or fall out from it.
市场效率作为一个假设交易者策略对价格时间序列产生集体预测结果的广泛概念，被 [Timmermann 和 Granger, 2004] 扩展和调整。在这项工作中，作者引入了市场效率的模型特定定义，即价格变化是独立的且不可预测的，条件是交易者使用的预测模型将可用信息映射为预测：在有效市场中，存在一组模型，其未来贴现回报的期望为零。此外，作者引入了市场效率的时间局部性概念，即市场可能暂时无效然后变得有效，或者反之。事实上，市场参与者及其交易策略的演化动态反映了市场效率的适应性 [Lo, 2005]，并提出了建模非均衡市场动态的挑战，即金融市场如何接近效率或偏离效率。
In [Lim and Brooks, 2011], the authors observe that the possibility of weak-form time-varying market efficiency has received increasing attention in the last decades, showing as evidence the time-varying structure of auto correlations measured in [Ito and Sugiyama, 2009]. Although accounting for model specification, the model of market efficiency by [Timmermann and Granger, 2004] is effectively limited by the need to introduce an arbitrary pricing kernel (viz. pricing model) to actually conduct a forecasting test over the returns. The need to assume a specific asset pricing model, the so-called “joint hypothesis problem”, was tackled by the seminal paper, [Jarrow and Larsson, 2012], where the authors combine the definition of an economy, consisting of a set of investors trading on a market, and define market efficiency as the ability of the financial market price to converge towards an equilibrium price process, thus avoiding the need to impose an asset pricing model.
在 [Lim and Brooks, 2011] 中，作者观察到弱式时变市场有效性的可能性在过去几十年中受到了越来越多的关注，并以 [Ito and Sugiyama, 2009] 中测量的自相关时变结构作为证据。尽管考虑了模型设定，[Timmermann and Granger, 2004] 的市场有效性模型实际上受到引入任意定价核（即定价模型）以实际对收益进行预测测试的需求的限制。假设特定资产定价模型的需求，即所谓的“联合假设问题”，在 [Jarrow and Larsson, 2012] 的开创性论文中得到了解决，作者结合了由一组在市场交易的投资者组成的经济体的定义，并将市场有效性定义为金融市场价格向均衡价格过程收敛的能力，从而避免了强加资产定价模型的需求。
Real markets cannot be regarded as such efficient aggregator s of information that no trader should ever be so naive as to hope to beat them. Traders make mistakes, markets are not perfect, and cannot be regarded as strongly efficient, as noted early in the literature, [Gordon and Kornhauser, 1985] and stated Fama himself, ”We would not, of course, expect this model to be an exact description of reality, and indeed, the preceding discussions have already indicated the existence of contradictory evidence.” However, experience reveals that even if markets are less efficient than the strong form of the hypothesis supposes, it is still very difficult to consistently beat them, which is why so few traders can allegedly do it. Therefore, we can consider these definitions as ideal scenarios serving as a framework for developing tests for quantifying real market efficiency.
现实市场不能被视为如此高效的信息聚合器，以至于没有交易者应该天真到希望击败它们。正如早期文献 [Gordon and Kornhauser, 1985] 所指出的，交易者会犯错，市场并不完美，也不能被视为强有效的。Fama 本人也指出：“我们当然不会期望这个模型是现实的精确描述，事实上，前面的讨论已经表明存在矛盾的证据。”然而，经验表明，即使市场的效率低于强有效假说所假设的水平，持续击败市场仍然非常困难，这也是为什么据说只有极少数交易者能够做到这一点。因此，我们可以将这些定义视为理想情景，作为开发量化现实市场效率测试的框架。
The notion of market efficiency as prices fully reflecting information can be interpreted in two main ways: (i) statistical unpredictability of the price time series and (ii) prices ought to be adherent to a fundamental value that the available information should entail [Black, 1986].
市场效率的概念，即价格完全反映信息，可以从两个主要方面来理解：(i) 价格时间序列的统计不可预测性，以及 (ii) 价格应当遵循由可用信息所蕴含的基本价值 [Black, 1986]。
These two interpretations, the statistical and the fundamental one, differ substantially in the ways they can be tested: whilst predictability of price time series can be formulated in terms of expectation over future profits, potentially including discount factors, dividends and pricing kernels [Walter, 2006, Timmermann and Granger, 2004], adherence to a fundamental value supposes the ability to build a structural model for the price formation based on the information available to traders.
这两种解释，即统计解释和基本面解释，在测试方式上有显著差异：虽然价格时间序列的可预测性可以用对未来利润的期望来表述，可能包括折现因子、股息和定价核 [Walter, 2006, Timmermann and Granger, 2004]，而遵循基本面价值则假设能够基于交易者可获得的信息构建价格形成的结构模型。
Measuring efficiency based on the discrepancy between asset prices and their fundamental values is difficult for one main reason: even if there were a structural model for the value of an asset, it would need to take into account the actual market where the asset is traded and the prices that traders would be willing to pay for it, hence it would need to model the market itself. Un surprisingly, tests for market efficiency have focused on the statistical interpretation of the efficient market hypothesis, mainly testing the ability of factor models including available information to predict price returns, [Fama et al., 1969, Fama and French, 1993]. This major limitation was addressed in [Jarrow and Larsson, 2012], where the authors introduce an economy associated with the market and define market efficiency as the correspondence between the security market price process and the equilibrium price process, [Duffie, 1986], of a commodity economy. The authors avoid the “joint hypothesis problem”: the necessity of testing market efficiency within a given equilibrium model. Their definition reconciles the statistical and fundamental interpretation of market efficiency by showing the correspondence between the notion of no-arbitrage, [Delbaen and Sch a cher mayer, 1994], in the security market, and the existence of an equilibrium price process for the underlying commodity market. We will mainly use the formalism and equilibrium notions introduced in [Jarrow and Larsson, 2012] to discuss the out-of-equilibrium evolution of market (in)efficiency.
基于资产价格与其基本价值之间的差异来衡量效率存在一个主要困难：即使存在资产价值的结构模型，它也需要考虑资产交易的实际市场以及交易者愿意支付的价格，因此需要建模市场本身。不出所料，市场效率的测试主要集中在有效市场假说的统计解释上，主要测试包括可用信息的因子模型预测价格回报的能力 [Fama et al., 1969, Fama and French, 1993]。这一主要限制在 [Jarrow and Larsson, 2012] 中得到了解决，作者引入了与市场相关的经济，并将市场效率定义为证券市场价格过程与商品经济的均衡价格过程 [Duffie, 1986] 之间的对应关系。作者避免了“联合假设问题”：在给定均衡模型中测试市场效率的必要性。他们的定义通过展示证券市场中的无套利概念 [Delbaen and Sch a cher mayer, 1994] 与基础商品市场均衡价格过程的存在之间的对应关系，调和了市场效率的统计和基本解释。我们将主要使用 [Jarrow and Larsson, 2012] 中引入的形式主义和均衡概念来讨论市场（非）效率的非均衡演化。
3 Market efifciency, investors’ beliefs, and trading strategies
3 市场效率、投资者信念与交易策略
The efficient market hypothesis models efficiency in terms of the statistical unpredictability of the return time series that derives from the necessity of the absence of arbitrage opportunities. It starts from an idealised view of financial markets where traders have access to the same information set, [Fama, 1970], and from that it hypothesizes that markets with informed traders should display randomly fluctuating returns with no arbitrage opport unities, or, more generally, efficiency should imply the existence of an equivalent martingale measure for the price process.
有效市场假说从统计上不可预测的回报时间序列的角度来建模市场效率，这种不可预测性源于不存在套利机会的必要性。该假说始于对金融市场的理想化观点，即交易者可以访问相同的信息集 [Fama, 1970]，并由此假设，拥有知情交易者的市场应表现出随机波动的回报，且不存在套利机会，或者更一般地说，效率应意味着价格过程存在一个等价的鞅测度。
The information set is a cumulative collection of time-structured data, including traditional macroeconomic time series, financial indicators, alternative data, high-resolution market data, fundamental data about individual stocks, firms, sectors, and more. In [Jarrow and Larsson, 2012] the information available to traders is expressed as a filtration over a probability space. The investors’ beliefs in [Jarrow and Larsson, 2012] represent the level of model capacity the investors have achieved, i.e. they constitute a form of market intelligence. In this definition of market efficiency, investors’ beliefs are crucial as they need to be equivalent to the measure of the probability space. In [Jarrow and Larsson, 2012] the probability measure defines the price process, which is defined as efficient if it corresponds to the equilibrium price process of a security and commodity market. In this setting, model complexity is embedded in the probability measure and market efficiency implies the existence of an equivalent martingale measure. Both these generalised definitions allows us to consider market (in)efficiency not only at a global market level but at a trader’s level. In particular, in [Jarrow and Larsson, 2012], the economy associated to the market consists of a finite set of investors each provided with their beliefs, information, and utility functions. Beliefs, information, and utility functions all contribute to shape the individual investor’s problem and consequently their optimal consumption and trading strategy. In this framework, in presence of market efficiency, all the individual optimal trading strategies are maximal, as demonstrated in [Jarrow and Larsson, 2012], and, notably, are dominated by a simple market portfolio (and by each security holding). Given any market, to disprove efficiency one simply needs to find an arbitrage opportunity or a trading strategy dominating the market portfolio. Arbitrage opportunities can emerge when the assumptions that lead to a market equilibrium are not met and investors fail to identify an optimal solution to the investor’s problem, so that their consumption choices and the trading strategies are not optimal, and the market prices do not necessarily provide an equilibrium price process for the economy.
信息集是一个累积的时间结构化数据集合，包括传统的宏观经济时间序列、金融指标、替代数据、高分辨率市场数据、个股、公司、行业的基本面数据等。在 [Jarrow and Larsson, 2012] 中，交易者可获得的信息被表示为概率空间上的一个过滤。在 [Jarrow and Larsson, 2012] 中，投资者的信念代表了他们达到的模型能力水平，即它们构成了一种市场智能。在这种市场效率的定义中，投资者的信念至关重要，因为它们需要与概率空间的测度等价。在 [Jarrow and Larsson, 2012] 中，概率测度定义了价格过程，如果它对应于证券和商品市场的均衡价格过程，则被定义为有效。在这种设定中，模型复杂性嵌入在概率测度中，市场效率意味着存在一个等价的鞅测度。这两种广义定义使我们不仅能在全球市场层面考虑市场（非）效率，还能在交易者层面考虑。特别是在 [Jarrow and Larsson, 2012] 中，与市场相关的经济由一组有限的投资者组成，每个投资者都有自己的信念、信息和效用函数。信念、信息和效用函数共同塑造了个体投资者的问题，从而影响了他们的最优消费和交易策略。在这个框架中，在市场效率存在的情况下，所有个体的最优交易策略都是最大的，正如 [Jarrow and Larsson, 2012] 中所证明的那样，并且值得注意的是，它们被一个简单的市场组合（以及每个证券持有）所主导。对于任何市场，要证明其无效性，只需找到一个套利机会或一个优于市场组合的交易策略。当导致市场均衡的假设不成立且投资者未能找到投资者问题的最优解时，套利机会可能会出现，因此他们的消费选择和交易策略不是最优的，市场价格不一定为经济提供均衡价格过程。
3.1 From efifcient markets to efifcient traders
3.1 从有效市场到有效交易者
The efficient market in [Jarrow and Larsson, 2012] is associated with an equilibrium price process which embodies the structural model that should map all the available information into prices; all traders have access to the available information and their beliefs are equivalent to the price probability measure. In real markets, traders differ based both on the information they can gather, their beliefs’ and on the trading strategies they can build. Formally, we have two separate objects: the complete filtered probability space of the market, and the investors’ beliefs with their information.
[Jarrow and Larsson, 2012] 中的有效市场与一个均衡价格过程相关联，该过程体现了将所有可用信息映射到价格中的结构模型；所有交易者都能获取可用信息，并且他们的信念等同于价格概率测度。在现实市场中，交易者基于他们能够收集的信息、他们的信念以及他们能够构建的交易策略而有所不同。形式上，我们有两个独立的对象：市场的完整过滤概率空间，以及投资者的信念及其信息。
We then proceed to classify different investors in the market based on their relative ability to detect signal in the market, i.e. to align their beliefs to the probability measure of the information set. In practice, this is an inference problem for traders who need to update their beliefs based on the observed information and price processes.
我们接着根据市场中不同投资者检测信号的能力，即将其信念与信息集的概率测度对齐的能力，对他们进行分类。在实践中，这对交易者来说是一个推断问题，他们需要根据观察到的信息和价格过程来更新自己的信念。
Let us assume a market where the set of available information is the same for all traders, and it coincides with the set of information of the market itself. We can now consider two main cases:
让我们假设一个市场，其中所有交易者可获得的信息集是相同的，并且与市场本身的信息集一致。我们现在可以考虑两种主要情况：
In this scenario, there is no guarantee for market efficiency, and as a consequence the no-dominance does not apply, i.e. there could be profitable admissible strategies that outperform the market portfolio, and we expect the efficient traders to be the ones to identify them.
在这种情况下，市场效率无法得到保证，因此无支配性不适用，即可能存在优于市场组合的可盈利且可接受的策略，我们预计高效的交易者将是识别这些策略的人。
Efficient traders need both to identify the right probability measure and the optimal solution to the investor’s problem. This comes at a cost, that we name optimization cost, defined as the cost needed to discover the probability measure, and, even if the true probability measure is available to efficient traders, they still may not be able to identify the optimal strategies. If only some traders succeed, then the market is inefficient, there exists a non-empty set of traders who can make predictably profitable trades, i.e. the market can exhibit arbitrage opportunities. In inefficient markets, efficient traders can beat the collective intelligence of the market. For a simple isolated small slow human investor using limited information, i.e. an uninformed simple trader, in an inefficient market, it should be almost impossible to do better than efficient traders. In inefficient markets competition is key and no trader can simply assume that a simple market portfolio will not be outperformed (on average) by better strategies. The emergence of market efficiency is positive especially for uninformed traders, i.e., when market efficiency applies, uninformed traders can simply invest in the market portfolio and forget about complex tasks such as aligning their beliefs and identifying complex (and expensive) trading strategies. This is a paradoxical perspective provided by market efficiency: it eliminates the out-of-equilibrium dynamic competition that is the very reason market efficiency should emerge.
高效的交易者需要同时识别正确的概率度量并找到投资者问题的最优解。这需要付出成本，我们称之为优化成本，即发现概率度量所需的成本。即使高效的交易者可以获得真实的概率度量，他们仍可能无法识别出最优策略。如果只有部分交易者成功，那么市场就是低效的，存在一组非空的交易者可以做出可预测的盈利交易，即市场可能表现出套利机会。在低效市场中，高效的交易者可以击败市场的集体智慧。对于一个使用有限信息的孤立、小型、缓慢的人类投资者（即无知的简单交易者）来说，在低效市场中几乎不可能比高效交易者做得更好。在低效市场中，竞争是关键，没有交易者可以简单地假设一个简单的市场组合不会被更好的策略（平均而言）超越。市场效率的出现尤其对无知的交易者有利，即当市场效率适用时，无知的交易者可以简单地投资于市场组合，而无需考虑诸如调整其信念和识别复杂（且昂贵）的交易策略等复杂任务。这是市场效率提供的一个悖论视角：它消除了非均衡动态竞争，而这正是市场效率应该出现的原因。
3.2 Co-evolution of price and beliefs
3.2 价格与信念的共同演化
In real (in)efficient markets, we still expect arbitrage opportunities to disappear, i.e. inefficient markets should approach efficiency: we expect the distance between the equilibrium price process and the realised price process to remain small. As discussed in [Timmermann and Granger, 2004], ”An efficient market is thus a market in which predictability of asset returns, after adjusting for time-varying risk-premia and transaction costs, can still exist but only ‘locally in time’ in the sense that once predictable patterns are discovered by a wide group of investors, they will rapidly disappear through these investors’ transactions.” In other words, predictable patterns, when they exist, they tend to self- destruct after a certain period of time. Arguably, this is how the collective intelligence of markets should emerge: the more traders expand their information sets and improve their capacity of estimating beliefs, the more the set of probability measure should evolve and push the equilibrium price process to higher complexity levels.
在实际的（非）有效市场中，我们仍然期望套利机会消失，即非有效市场应趋向于有效：我们期望均衡价格过程与实际价格过程之间的距离保持较小。正如 [Timmermann and Granger, 2004] 所讨论的，“一个有效市场是一个在调整了时变风险溢价和交易成本后，资产回报的可预测性仍然存在但仅在‘时间上局部’存在的市场，即一旦可预测的模式被广泛投资者发现，这些模式将通过投资者的交易迅速消失。” 换句话说，可预测的模式在存在时，往往会在一定时间后自我毁灭。可以说，这是市场集体智慧应如何显现的方式：交易者扩展其信息集并提高其估计信念的能力越强，概率测度集就越应演变，并将均衡价格过程推向更高的复杂性水平。
The presence of an arbitrage opportunity could be associated to information asymmetry and to the ability of some investors to better identify optimal trading strategies thanks to their better information gathering and belief estimation. This fundamental mechanism creates a pressure for the time evolution of the equilibrium price process in efficient markets: the market price cannot be in different to investors’ beliefs even when they are potentially far from the market probability measure as this would lead to the persistence of arbitrage opportunities. This could be the case if investors were systematically unable to gather relevant information for the price process or to approximate the probability measure, due to modeling limitations. In this setting, for the market to be efficient it would need to temporarily co-evolve with the traders’ beliefs, i.e. the probability measure would need to be aligned with the investors’ beliefs and not the other way around, at least until traders were able to update their information and improve their modeling capacity. In particular, here we argue that the probability measure complexity should co-evolve with the complexity of the investors’ beliefs in the market. In an evolving close-to-efficiency inefficient market, the probability measure should evolve over time, so that its complexity remains close but above the average complexity of the beliefs in the market.
套利机会的存在可能与信息不对称以及某些投资者由于其更好的信息收集和信念估计能力而能够更好地识别最佳交易策略有关。这种基本机制对有效市场中均衡价格过程的时间演化产生了压力：市场价格不能对投资者的信念漠不关心，即使这些信念可能与市场概率测度相差甚远，因为这会导致套利机会的持续存在。如果投资者由于建模限制而系统性地无法收集与价格过程相关的信息或近似概率测度，则可能会出现这种情况。在这种情况下，为了使市场保持有效，它需要暂时与交易者的信念共同演化，即概率测度需要与投资者的信念保持一致，而不是相反，至少在交易者能够更新其信息并提高其建模能力之前。特别是，我们认为概率测度的复杂性应与市场中投资者信念的复杂性共同演化。在一个接近有效但尚未完全有效的市场中，概率测度应随时间演化，使其复杂性保持接近但高于市场中信念的平均复杂性。
So far we have considered traders with different beliefs’ but sharing the same information. In general, as originally discussed in [Fama, 1970], we can distinguish different information sets, for example market information on price time series, public information on companies, and private information on business strategies and more. The acquisition and integration of information in the models available to traders and in equilibrium price process are a fundamental part of the market evolution. In [Jarrow and Larsson, 2012], the cases of information reduction and information expansion are considered: information reduction, namely the reduction of the filtration in the filtered probability space, is found to preserve market efficiency whilst information expansion is generally not. The probability measure may be based on a large information set and traders may have access to limited information sets that hinder their ability to design profitable strategies. Conversely, the acquisition of new information by traders could lead to the identification of new equilibrium price processes depending on larger information sets. The set of data that are now available about companies and their time resolution has significantly increased and, at the same time, this requires an increase in the models’ ability to incorporate this information and translate that into optimal predictions.
到目前为止，我们考虑了具有不同信念但共享相同信息的交易者。一般来说，正如[Fama, 1970]最初讨论的那样，我们可以区分不同的信息集，例如价格时间序列的市场信息、公司的公共信息以及商业策略等的私人信息。交易者可用的模型和均衡价格过程中的信息获取与整合是市场演化的基本部分。在[Jarrow and Larsson, 2012]中，考虑了信息减少和信息扩展的情况：信息减少，即过滤概率空间中过滤的减少，被发现可以保持市场效率，而信息扩展通常不能。概率测度可能基于一个大的信息集，而交易者可能只能访问有限的信息集，这阻碍了他们设计盈利策略的能力。相反，交易者获取新信息可能会导致识别出依赖于更大信息集的新均衡价格过程。现在可用的关于公司及其时间分辨率的数据集显著增加，同时，这也要求模型的能力增强，以便能够整合这些信息并将其转化为最优预测。
The cost of collecting and processing this increasing amount of data pushes traders to improve their ability of extracting signal from them, i.e. a large investment on data collection needs to be coupled with an investment in the model complexity, at least up to a certain point, to make sure that all the trading signal that is present in the data is identified and exploited. This implies that the investors’ beliefs and the equilibrium price process should co-evolve with the set of available information. More information requires more complex models, able to extract new signals from the new information; conversely the acquisition of new information pushes traders to improve their beliefs’, which as a consequence should push the equilibrium price process to higher complexity levels.
收集和处理这些日益增长的数据的成本促使交易者提高从中提取信号的能力，即对数据收集的大量投资需要与模型复杂性的投资相结合，至少在一定程度上，以确保数据中存在的所有交易信号被识别和利用。这意味着投资者的信念和均衡价格过程应该与可用信息集共同演化。更多的信息需要更复杂的模型，能够从新信息中提取新的信号；反过来，新信息的获取促使交易者改进他们的信念，这反过来应该推动均衡价格过程达到更高的复杂性水平。
4 Low-cost universal ap proxima tors as drivers of market efifciency
4 低成本通用近似器作为市场效率的驱动因素
The time evolution of market efficiency has been investigated mainly through the study of the existence of regime changes and structural changes in auto regressive models, [Lim and Brooks, 2011]. Linear models have the advantage of efficient and reliable solvers to identify the optimal models, but only consider linear combinations of factors to determine the price. Universal ap proxima tors have been known and studied in functional analysis for a long time, [Stone, 1948], and recently multilayer perce ptr on s, [Hornik et al., 1989, Barron, 1993], and their generalisations, have been recognised and widely utilised as such. In particular, in [Hornik et al., 1989], the authors demonstrated that multilayer feed forward networks are capable of approximating any Borel-measurable function from one finite dimensional space to another to any desired degree of accuracy, provided sufficiently many hidden units are available, and multiple other studies have extended and generalised this result, [Cybenko, 1989, Barron, 1993].
市场效率的时间演变主要通过研究自回归模型中制度变化和结构变化的存在来进行探讨 [Lim and Brooks, 2011]。线性模型的优势在于能够通过高效且可靠的求解器识别最优模型，但仅考虑因素的线性组合来确定价格。通用逼近器在函数分析中早已被认识和研究 [Stone, 1948]，最近多层感知器 [Hornik et al., 1989, Barron, 1993] 及其推广形式已被广泛认可和利用。特别是在 [Hornik et al., 1989] 中，作者证明了多层前馈网络能够以任意精度逼近从一个有限维空间到另一个有限维空间的任何Borel可测函数，前提是有足够多的隐藏单元可用，并且多项其他研究扩展和推广了这一结果 [Cybenko, 1989, Barron, 1993]。
The ability of approximating a function with a class of other basis functions is not a novel result, yet the limited generalisation errors and the efficiency of the training algorithms, yielding relatively low computational costs, for neural networks, [Rumelhart et al., 1986], have notoriously established the use of these models in a wide range of applications. In finance, a growing number of studies is showing how AI algorithms can outperform traditional statistical models, e.g. [Fischer and Krauss, 2018, Makridakis et al., 2020], and multiple independent surveys on financial firms have recognised that a vast majority of traders and investors, [Jung et al., 2 will be relying more and more on AI models for their financial decisions. The ability of AI models of acting as universal function ap proxima tors, also in presence of different modalities of data, such as text, time-series, images, makes them suitable models to adopt in the space of available models for traders. Despite the existence of other universal approximators, AI models are unique in terms of multi modality, capacity, and adaptability. Linear ap proxima tors are generally not multimodal, not easy to extend in terms of model capacity (unless they take the form of AI models themselves), and less flexible in terms of available architectures and optimisation schemes. Universal approximation theorems guarantee that, provided the convergence of the training algorithm, with sufficient data all informed AI traders will never under-perform other traders and, if existent, should always be able to identify and approximate the probability measure of the equilibrium price process. We argue that, under these hypotheses, a market of rational traders will be inclined to adopt AI models and, as discussed above, the equilibrium price process should be pressured to become AI-efficient, i.e. not even AI models should, eventually, be able to identify profitable strategies. We may refer to this as asymptotic-AI-efficiency. At that stage, the competition in the market will focus on the rapid integration of new information, rather than on the model itself. Convergence and big data are not guaranteed, so models will keep changing in terms of data available and in terms of players in the market.
用一类其他基函数近似函数的能力并不是一个新结果，然而神经网络有限的泛化误差和训练算法的高效性，使得计算成本相对较低，[Rumelhart et al., 1986]，这些模型在广泛的应用中得到了广泛使用。在金融领域，越来越多的研究表明，AI算法可以超越传统的统计模型，例如 [Fischer and Krauss, 2018, Makridakis et al., 2020]，并且多项针对金融公司的独立调查已经认识到，绝大多数交易员和投资者 [Jung et al., 2] 将在他们的金融决策中越来越依赖AI模型。AI模型作为通用函数近似器的能力，即使在面对不同模态的数据（如文本、时间序列、图像）时，也使得它们成为交易员在可用模型空间中的合适选择。尽管存在其他通用近似器，AI模型在多模态性、容量和适应性方面是独一无二的。线性近似器通常不具备多模态性，不易扩展模型容量（除非它们本身采用AI模型的形式），并且在可用架构和优化方案方面灵活性较低。通用近似定理保证，只要训练算法收敛，并且有足够的数据，所有知情的AI交易员永远不会表现逊色于其他交易员，并且如果存在，应该始终能够识别并近似均衡价格过程的概率测度。我们认为，在这些假设下，理性交易员的市场将倾向于采用AI模型，并且如上所述，均衡价格过程将被迫成为AI有效的，即最终即使是AI模型也不应该能够识别出有利可图的策略。我们可以将此称为渐近AI有效性。在那个阶段，市场的竞争将集中在快速整合新信息上，而不是模型本身。收敛性和大数据并不能保证，因此模型将在可用数据和市场参与者方面不断变化。
4.1 Trading in a world of AI-efifcient traders
4.1 在AI高效交易者世界中的交易
The notion of economy-dependent market efficiency, [Jarrow and Larsson, 2012], is extremely useful to provide a richer picture of economic equilibrium in financial markets, yet it leaves us with complex questions over markets’ evolution: how does the space of available information evolve? And, how do beliefs and price process co-evolve?
经济依赖型市场效率的概念 [Jarrow and Larsson, 2012] 在提供金融市场经济均衡的更丰富图景方面极为有用，但它也给我们留下了关于市场演化的复杂问题：可用信息的空间如何演化？以及，信念和价格过程如何共同演化？
Hardware and software design has led to an increased capability to harness information and to an increased ability to explore the space of statistical models for estimating beliefs and developing trading strategies. The specific way in which the price process varies over time depends on the market fundamentals, market design, market micro structure, e.g. the liquidity of the order book, and ultimately on the traders themselves.
硬件和软件设计的进步提升了我们利用信息的能力，也增强了我们探索统计模型空间的能力，这些模型用于估计信念和开发交易策略。价格过程随时间变化的具体方式取决于市场基本面、市场设计、市场微观结构（例如订单簿的流动性），最终取决于交易者本身。
In a realistic setting, traders share a certain fraction of information, as well as certain classes of models, nevertheless it is also likely that each trader will have a slightly -or significantly- different set of information and, at the same time, a different class of models that they are able to use. Hence, we should look at the changes in market (in)efficiency in two main directions: one is a longitudinal time dimension, i.e. information and beliefs change and evolve over time; and another one is cross-sectional, i.e. in a given market at a given time multiple traders can coexist who do not share the same information and the same beliefs.
在实际环境中，交易者共享一定比例的信息以及某些类别的模型，但每个交易者也可能拥有略微或显著不同的信息集，同时能够使用不同类别的模型。因此，我们应该从两个主要方向来观察市场（非）效率的变化：一个是纵向时间维度，即信息和信念随时间变化和演变；另一个是横截面维度，即在给定时间的给定市场中，可能存在多个交易者，他们不共享相同的信息和信念。
The absence of profitable trades is hypothesized as a consequence of the market ability to identify perfect prices based on the information available to different traders, and this efficiency is enabled by the complexity of the equilibrium price process that is accounting for the collective intelligence of the market.
假设无利可图的交易是市场基于不同交易者可获得的信息识别完美价格的能力的结果，这种效率是由均衡价格过程的复杂性所实现的，该过程考虑了市场的集体智慧。
So far we have made the simplifying assumption that the equilibrium price process at a given time is a function of the information set which is independent from the beliefs and strategies of individual traders. In reality markets do display some form of co-dependence and reflexivity: the performance of a strategy will not only be based on its expected value but also on the simultaneous strategies of the other traders, e.g. the specific sequence in order placement can significantly affect the price that a trader can obtain with respect to others. The dynamic time-dependent mapping between information and optimal trades will depend on the simultaneous, or quasi-simultaneous, decisions of the other traders, that will partially move the optimal solution with each of their trades. This can be partially understood in a game-theoretical way. A well-known example is the following. Consider a game where players (traders) must guess a number between 0 and 100 and the winner is the player who guesses the number which is closest to $2/3$ of the average guess. The Nash equilibrium, i.e. the solution to this game, is 0, so in theory it is impossible to beat the crowd (the market) if every player uses the optimal strategy. But real markets are not perfect as we said. In fact, this contest was actually run by the Financial Times in 1997 and the result was that, although many contestants were able to figure out the Nash equilibrium and guessed 0, they were wrong in thinking that everyone else would be as smart as they were, [Thaler, 2016]. In fact the average guess was 18.9 and so the winning guess was 13. This experiment thus shows that the performance of a strategy in a real-market does indeed depends on the simultaneous strategies of the other traders and so to beat the market a trader must model also all other traders’ behavior.
到目前为止，我们做了一个简化的假设，即给定时间的均衡价格过程是信息集的函数，而该信息集与个体交易者的信念和策略无关。实际上，市场确实表现出某种形式的相互依赖性和反身性：一个策略的表现不仅基于其预期价值，还基于其他交易者的同时策略，例如，订单提交的特定顺序会显著影响交易者相对于其他人能够获得的价格。信息与最优交易之间的动态时间依赖映射将取决于其他交易者的同时或准同时决策，这些决策会随着他们的每一次交易部分地移动最优解。这可以从博弈论的角度部分理解。一个著名的例子如下：考虑一个游戏，玩家（交易者）必须猜测一个介于0到100之间的数字，获胜者是猜测最接近平均猜测值的$2/3$的玩家。纳什均衡，即这个游戏的解，是0，因此理论上，如果每个玩家都使用最优策略，就不可能击败群体（市场）。但正如我们所说，真实市场并不完美。事实上，这个比赛实际上是由《金融时报》在1997年进行的，结果是，尽管许多参赛者能够计算出纳什均衡并猜测0，但他们错误地认为其他人也会像他们一样聪明 [Thaler, 2016]。实际上，平均猜测值是18.9，因此获胜的猜测值是13。这个实验表明，真实市场中策略的表现确实取决于其他交易者的同时策略，因此要击败市场，交易者还必须建模所有其他交易者的行为。
In the framework we presented for modeling efficiency, we essentially have two rate of changes in complexity, the change in market complexity, that is how rapidly the complexity of the probability measure changes over time and how rapidly the complexity of the beliefs’ models available to traders change.
在我们提出的建模效率框架中，本质上存在两种复杂性的变化率：市场复杂性的变化，即概率度量的复杂性随时间变化的速度，以及交易者可用的信念模型的复杂性变化的速度。
Traders’ directly affect market efficiency and they compete with the market price dynamics they actually contribute to define. Market efficiency is related to the models available to the traders who populate it. Let us, for instance, consider a market populated with traders with limited information and limited model capacity. It is likely that a new trader with more information and more model capacity could enter the market and easily find profitable strategies, i.e. the market would be inefficient. The precise mechanism by which the traders’ ability is translated into the collective intelligence of the market is generally dependent on the details of the market design.
交易者直接影响市场效率，他们与市场价格动态竞争，而他们实际上参与了这些动态的定义。市场效率与市场中交易者可用的模型有关。例如，让我们考虑一个由信息有限和模型能力有限的交易者组成的市场。一个拥有更多信息和更强模型能力的新交易者可能会进入市场，并轻松找到盈利策略，即市场将是低效的。交易者能力转化为市场集体智慧的具体机制通常取决于市场设计的细节。
This complex adaptation makes financial markets different from a static and predictable computational task, such as classifying pictures or even solving combinatorial optimisation problems, and constitutes the very reason a market of investments or hedge funds exists: profitable trades are the consequence of a powerful prediction model and of a successful behavioral agent-based model [Bouchaud, 2018]. Traders are both responsible for the emergence of the collective intelligence of the markets whilst being in competition with it and, as agents, are effectively following a noisy learning algorithm trying to optimise their different objective functions in a non-stationary multi-player environment.
这种复杂的适应性使得金融市场不同于静态且可预测的计算任务，例如图片分类甚至解决组合优化问题，这也是投资市场或对冲基金存在的根本原因：盈利交易是强大预测模型和成功的行为智能体模型的结果 [Bouchaud, 2018]。交易者既是市场集体智慧涌现的推动者，又与之竞争，作为智能体，他们实际上是在一个非稳态的多玩家环境中遵循一种嘈杂的学习算法，试图优化各自的目标函数。
5 Conclusions
5 结论
In this paper, we discussed the notion of complexity in market efficiency and the importance of its out-of-equilibrum dynamics. Introducing different levels of complexity can allow market observers to describe the evolution of market (in)efficiency over time and make sense of the ability of traders to adapt to changing market conditions. More complex models are now available to traders to map the information into beliefs and beliefs into trading strategies, potentially leading to a form of market singularity, i.e. where all traders can effectively learn accurate prices instantaneously. We discussed how the existence of universal ap proxima tors and their increasing availability and decreasing computational costs could affect the evolution of market efficiency. On one hand, low-cost AI ap proxima tors could push the boundary of market efficiency to a highly competitive global AI-efficiency. On the other hand, models are limited by convergence, computational power, and by the complex modeling interactions between strategies, whilst we are observing the complex if i cation of the economic dependencies, of the interconnected ness of countries, firms, and supply networks, the increase in the responsiveness of consumers’ preferences, which can all contribute to hinder the predictability of the market performance of different stocks and companies, so that even more information and more complex models will be needed to reach a given market performance, and markets will complexify at a rate not even AI-traders will be able to adapt to.
在本文中，我们讨论了市场效率中的复杂性概念及其非均衡动态的重要性。引入不同层次的复杂性可以使市场观察者描述市场（非）效率随时间的演变，并理解交易者适应不断变化的市场条件的能力。现在，交易者可以使用更复杂的模型将信息映射为信念，并将信念映射为交易策略，这可能导致一种市场奇点的形式，即所有交易者都能即时有效地学习准确的价格。我们讨论了通用近似器的存在及其可用性的增加和计算成本的降低如何影响市场效率的演变。一方面，低成本的AI近似器可能会将市场效率的边界推向高度竞争的全球AI效率。另一方面，模型受到收敛性、计算能力以及策略之间复杂建模交互的限制，同时我们观察到经济依赖性的复杂化、国家、公司和供应网络的互联性增加，消费者偏好的响应性增强，这些都可能阻碍不同股票和公司市场表现的可预测性，因此需要更多的信息和更复杂的模型才能达到给定的市场表现，而市场的复杂化速度甚至AI交易者也无法适应。
